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Questions tagged [potential-theory]

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

Questions in this tag will ideally be given at least one other tag, since potential theory lies on the intersection of fields. Some of the possibly appropriate tags: , , , , , , and .

354 questions
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Find the Green function of the upper half ball by using the Green function of the whole ball

Find the Green function of the upper half ball $\Omega:=\left\{x\in\mathbb{R}^n|\lVert x\rVert0\right\}$ (for the Dirichlet boundary value problem of the Laplace equation). Show that the function you found is indeed a Green function. HINT:…
user34632
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vote
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Is solution of Laplace's equation square integrable

I am wondering if the solution of Laplace's equation (e.g. electrostatic potential $\Phi$ which satisfies $\nabla^2\Phi=0$) is square-integrable? I am confused, because in one dimension in the large distance limit, $\Phi(x)\sim\frac{1}{x}$. So…
kolahalb
  • 91
0
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Two more little tasks concerning Dirichlet boundary value problem

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap C^{0,\lambda}(B_R(0))$ and $0<\lambda<1$. (a) Give the…
user34632
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0 answers

Complex potential/ observable

I have an observable denoted by $C$, related to a complex potential $B$ by : $$ C= \bar{B}B ,$$ where $B$ is a complex potential. I know that $ \left. C \right|_0 =C_0 $, a known constant, where the evaluation at $_0$ denotes an equilibrium \…
yourlazyphysicist
  • 481
0
votes
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Velocity potential in conical coordinate

Velocity potential in conical coordinate I just want to make sure that the equation regarding velocity potential in conical coordinates (the above picture) is correct or if there is a typo. I came across this equation in an ISI paper and have been…
Amin Azimi
  • 1
0
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How to show that $\int_{B(x, t)} \Delta f d \lambda=t^{N-1} \int_{S} \frac{\partial}{\partial t} f(x+t y) d \sigma(y)$?

Let $f$ be a $C^{2}$ function on an open set which contains $\overline{B(x, r)} .$ We can use first Green's formula and then differentiation under the integral sign to obtain $$ \int_{B(x, t)} \Delta f d \lambda=\int_{S(x, t)} \frac{\partial…
BRH
  • 31
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Helmholtz equation Inequality (potential theory)

I was reading the chapter about surface potential. I came across an inequality $$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|,$$ $k$ is a complex number here. Since $k$ is a complex number here, I am a bit confused. I am wondering what is the…
000000000
  • 497
0
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1 answer

Logarithmic capacity invariant to modifcations by polar sets?

Let $E \subset \mathbb{C}$ be a compact set. Its logarithmic capacity is defined as $$ \DeclareMathOperator{\capacity}{cap} \capacity(E) := \exp\left(-\inf_\mu \iint \log\frac{1}{|z-w|} \, d\mu(z) \, d\mu(w)\right), $$ where the infimum is taken…
gTcV
  • 671
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Phase Potentials

Looking to get a explanation of the following solution. $x'=sinx$ I understand that you must integrate $sinx$ and then take the negative of it making it $v(x)=cos(x) + c = cos(x)$ with $c=0$. However I am having problems understanding the…
mp12345
  • 113